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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>Note: Consider the case that <span class="process-math">\(\alpha\)</span> is a positive integer.(i) For <span class="process-math">\(\alpha=2 l,\)</span></p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
y_1=1+\sum_{k=1}^{\infty} (-1)^k \frac{\alpha (\alpha-2) \cdots (\alpha-2k+2) (\alpha+1)(\alpha+3)\cdots (\alpha+2k-1)}{(2k)!} x^{2k}.
\end{equation*}
</div>
<p class="continuation">Consider <span class="process-math">\(k=l+1\text{,}\)</span> then <span class="process-math">\(\alpha-2k+2=2 l - 2l=0\text{.}\)</span> For <span class="process-math">\(k \geq l+1\text{,}\)</span> all the coefficients will be zero. Therefore,</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
y_1=1+\sum_{k=1}^{l} (-1)^k \frac{\alpha (\alpha-2) \cdots (\alpha-2k+2) (\alpha+1)(\alpha+3)\cdots (\alpha+2k-1)}{(2k)!} x^{2k},
\end{equation*}
</div>
<p class="continuation">which is a polynomial of degree <span class="process-math">\(2l\text{.}\)</span>(ii) For <span class="process-math">\(\alpha=2l+1\text{,}\)</span></p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
y_2=x+\sum_{k=1}^{\infty} (-1)^k \frac{(\alpha-1)(\alpha-3)\cdots(\alpha-2k+1)(\alpha+2)(\alpha+4)\cdots(\alpha+2k)}{(2 k+1)!} x^{2k+1}.
\end{equation*}
</div>
<p class="continuation">Consider <span class="process-math">\(k=l+1\text{,}\)</span> <span class="process-math">\(\alpha-2k+1=0\text{.}\)</span> For <span class="process-math">\(k \geq l+1\text{,}\)</span> all the coefficients will be zero. Therefore,</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
y_2=x+\sum_{k=1}^{l} (-1)^k \frac{(\alpha-1)(\alpha-3)\cdots(\alpha-2k+1)(\alpha+2)(\alpha+4)\cdots(\alpha+2k)}{(2 k+1)!} x^{2k+1},
\end{equation*}
</div>
<p class="continuation">which is a polynomial of degree <span class="process-math">\(2l+1\text{.}\)</span></p>
<span class="incontext"><a href="sec5_3.html#p-224" class="internal">in-context</a></span>
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